I was having problems with my domain (routing to the wrong DNS) that nobody could fix, so in desperation I purchased a new domain and asked support staff to change my hosting account to the new domain.
I don't understand how your DNS routing couldn't be fixed. You should not have had to get a new domain name.
How did you copy over the files? If via FTP, sometimes FTP doesn't do the batch copy completely and files/folders can be missing. Make sure you have all the files/folders on the new server.
Did you upload appropriate files in ASCII?
Here's the User's Guide, so go through the Installation instructions and see if you're missing something.
http://www.phpbb.com/support/guide/